Mathematical Sciences: Geometric Group Theory and 3-Manifolds

群G关于一组 生成元A是其顶点是G的元素的图，并且 使得对于G中的每个g和A中的每个生成元a， 从G到Ga标记为发生器A。 这张图有 拓扑和几何性质，通常与 群G的性质，且与生成元的集合无关。 例如，格罗莫夫的“负弯曲”群的概念可以是 根据凯莱图的说法，有一个固定的 数c使得在凯莱中的每个最短路径三角形中 图中，三角形的每条边都在 其他两条边的结合。 还有其他各种 可以用类似的术语描述的属性，尽管 定义通常并不那么简单。 一个最 有趣的是斯蒂芬·G·砖，他 称之为“qsf”;给定一个组表示，一个构造2- 复形：以给定的群为基本群的复形 一个人问，是否任意大的社区， 通用封面中的身份可以被忠实地表示（在 有限1-连通复映射 如果是这样的话，那么这个群体被称为“准”。 简单过滤”或QSF。 有一些概括 Poenaru和Casson的结果，因此有一个定理， 如果一个非球面的，P2-不可约的，闭的3-流形具有 基本群是qsf，那么它的泛覆盖是 与R3同胚。 这些和相关的概念已经产生了革命， 无限群理论在过去的几年里。 基本 模式似乎是，人们发现一些事实，在微分 几何和拓扑的3流形;一个抽象的 基本群水平，然后发现群论 定理，有时有一个反向应用拓扑。 此外，受试者倾向于与 凯莱图本身的可计算性;托德-考克斯特 有限群的算法因此扩展到各种类型的 关于有限Cayley图的可计算性问题。 理论 “自动群”涉及有限状态自动机和正则 语言;然而，这个理论并不容易适用于某些语言。 幂零群和矩阵群等已知群， 整数;为了理解这些，需要扩展 有限状态机的概念在某些非常有限的方式 这还不明显。 也许这会对 形式语言理论和数学的其他方面通常 被称为“计算机科学”。"

The Cayley graph of a group G with respect to a set of generators A is the graph whose vertices are the elements of G, and such that for each g in G and each generator a in A, an edge runs from g to ga labeled with the generator a. This graph has topological and geometric properties which are often related to properties of the group G and independent of the set of generators. Gromov's notion of "negatively curved" group, for example, can be stated in terms of the Cayley graph as saying that there is a fixed number c such that in each triangle of shortest paths in the Cayley graph, each edge of the triangle is within the c-neighborhood of the union of the other two edges. There are various other properties which can be described in similar terms, although the definitions are not quite so simple, usually. One of the most interesting is a property invented by Stephen G. Brick which he calls "qsf"; given a group-presentation, one constructs the 2- complex which is described, having the given group as fundamental group; one asks whether arbitrarily large neighborhoods of the identity in the universal cover can be faithfully represented (in a certain technical sense) by finite, 1-connected complexes mapping into the universal cover; if so, the group is said to be "quasi- simply-filtrated" or qsf. There are generalizations of some results of Poenaru and Casson, so that one has a theorem to the effect that if an aspherical, P2-irreducible, closed 3-manifold has fundamental group which is qsf, then its universal cover is homeomorphic to R3. These and related notions have produced a revolution in the theory of infinite groups within the last few years. The basic pattern seems to be that one finds some facts in differential geometry and topology of 3-manifolds; one abstracts to the fundamental group level, and then one finds group-theoretic theorems which sometimes have a reverse application to topology. In addition, the subject tends to have a connection with the computability of the Cayley graph itself; the Todd-Coxeter algorithm for finite groups thus extends to various kinds of computability questions about finite Cayley graphs. The theory of "automatic groups" involves finite-state automata and regular languages; this theory, however, does not apply to certain easily understood groups such as nilpotent groups and matrix groups over the integers; in order to understand these, one needs to extend the notion of a finite-state machine in certain very restricted ways which are not yet obvious. Perhaps this will have implications in formal language theory and other aspects of mathematics usually considered to be "computer science."